Geometric probabilities for a cluster of needles and a lattice of rectangles

TL;DR

This paper studies the probability distribution of a cluster of needles intersecting a rectangular lattice, showing that as the number of needles increases, the distribution converges uniformly to a specific limit.

Contribution

It introduces a model for a cluster of needles on a rectangular lattice and proves the uniform convergence of the intersection distribution as the number of needles grows.

Findings

Distribution converges uniformly as number of needles increases

Limit distribution characterized for large clusters

Provides a probabilistic framework for needle-lattice interactions

Abstract

A cluster of $n$ needles ($1\leq n<\infty$) is dropped at random onto a plane lattice of rectangles. Each needle is fixed at one end in the cluster centre and can rotate independently about this centre. The distribution of the relative number of needles intersecting the lattice is shown to converge uniformly to the limit distribution as $n\rightarrow\infty$.